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5 Methods for Discrete Processing and Analysis of Biosignals

Fig. 5.17: Example of a discrete convolution according to section 5.3.1.3 calculated by a weighted

sum of shifted impulse responses.

The matrix cycl{gp} is cyclic. In contrast to linear convolution, the matrix elements of

the i-th column are also shifted downwards by one index with respect to the elements

of the i −1-th column, but the element of the i −1-th column that is omitted by the

downward shift is added again at the top of the i-th column, because there may only

be Np elements per column.

Explanatory Example

Let a periodically time-invariant system have, at a period length of Np = 3, the pulse

response

gp(0) = 1 ;

gp(1) = 0.75 ;

gp(2) = 0.5

and the periodic input signal

x(0) = 2 ;

x(1) = 1 ;

x(2) = 0.5 .